I am guessing you are talking about a car's HVAC system. If so, yes it will still work as the heating for your car comes from a heating coil fed from your engine coolant. There is a separate cooling coil fed from the AC system.
NO! THE HEATER CORE IS YOUR CARS HEAT SOURCE YOU CAN BYPASS BUT IS NOT A GOOD IDEAL IN WINTER!
yes but if the compressor does not work there is no reason for it.
yes
yes it will.
No electricity
It would be bypassed by the mode door in the ducting under the dash. You may have have a problem with your mode door if your compressor is turning on when you select A/C. If the compressor clutch does not engage it is likely that there is an insufficient charge in the system, and you may have a leak.
if the blower motor still works, somebody probably bypassed the heater core in the past.
Open circuit breakers. Poor electrical connections to the compressor, thermostat or bad wires not telling relay to cut the compressor on. Elaborate
because the compressor must be running to circulate the refrigerant in the system
Low refrigerant charge? Bad compressor?
For both your heater and air conditioning not to work means you either broke a belt or the belt is loose. Overheating would also be caused by this, because your water pump is driven by the same belt. Do not drive it until you get it fixed, you'll kill that engine by overheating it.
I am unsure what you mean by bypass. The Air Conditioning system will not work with out the compressor. The compressor can be disabled by unplugging the electric clutch. Most of these little Fords could have a different length belt fitted so the compressor pulley didn't turn at all.